# Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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### Flow of a vector field on a manifold with boundary

Let $(M,g)$ be a compact riemannian manifold with boundary, and let $N$ be the unit normal vector field defined on $\partial M$ and pointing outwards. Let $X$ be a smooth vector field defined on $M$, ...
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### Solutions for small perturbations

I have the following exercise: Let $f,g\colon \mathbb{R}\rightarrow \mathbb{R}$ be $C^1$ bounded functions and $f(0)=0$. Show that for suitably small $\varepsilon>0$ the following problem has a ...
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### Compute approximation of ODE using one step of explicit/implicit Euler method

I'm given the IVP: $$u^{(3)}(t) + u'(t) = tu(t)$$ $$u''(2) = 2$$ $$u(2) = 0$$ $$u'(2) = 1$$ and am asked to approximate the solution for $t=2.5$ using one step of the explicit Euler-method and one ...
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### Is it possible for the function to be a solution of this equation?

Consider the equation $$y′′ + p(t)y′ + q(t)y^3 = 0$$ with $p, q ∈ C(\mathbb{R}, \mathbb{R})$. Is it possible for the function $$y = e^t −\frac{t^2}{2} −t−1$$ to be a solution of this equation? I ...
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### Differential Equation, how to solve $(x^2-2xy-y^2)dy= (x^2+2xy-y^2) dx​$?

Solve$$(x^2-2xy-y^2)dy= (x^2+2xy-y^2) dx​$$ I've tried many methods but still unable to solve that problem
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### Show that $y(x)= \int_x^\infty e^{-t^2}dt$ satisfies the differential equation $y^{(2)}+2xy^{(1)}=0$. [on hold]

Show that $y(x)= \int_x^\infty e^{-t^2}dt$ satisfies the differential equation $y^{(2)}+2xy^{(1)}=0$.
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### $x(t)$ is a solution of the equation $x′ = f(t,x)$ such that $x(0) \neq 0$, show that $x(t) \neq 0,~ \forall t ∈ \mathbb{R}$.

Let's consider the following problem. Let $f(t,x) ∈ C(\mathbb{R} × \mathbb{R}^n,\mathbb{R}^n)$ and satisfy a local Lipschitz condition in $x$. Assume $f(t,0) = 0$. If $x(t)$ is a solution of the ...
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### Finding constant $a$ so that differential equation becomes correct

Question: Find the value of $a$ so that the function $$y = \sqrt{x} \ln{x}$$ is a solution to the differential equation $$y' - \frac{a}{x} \cdot y = \frac{1}{\sqrt{x}}$$ Attempted solution: My ...
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### Why Special Functions are called 'special'?

Why Special Functions are called 'special' ? What particular thing made it so special ?
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### Why the origin is globally asymptotically stable?

If the Lyapunov function is $$V(x) = x^2_1 + x^2_2-1$$ And its time derivative is $$\dot{V}(x) = -(x^2_1 + x^2_2)$$ Why the origin is globally asymptotically stable?
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### Difficult Second Order ODE solution?

I'm looking at the functional $$T[y] = \int_a^b\sqrt{\frac{1+y'^2}{2g(y-\mu x)}}\ \mathrm dx$$ and trying to minimize it via the Euler-Lagrange-equations, which I can do, however, it seems that the ...
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### How do we know $\sin$ and $\cos$ are the only solutions to $y'' = -y$?

According to Wikipedia, one way of defining the sine and cosine functions is as the solutions to the differential equation $y'' = -y$. How do we know that sin and cos (and linear combinations of them,...
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### A piecewise function as the output signal of an LTI system

In an LTI system, consider the following: The input signal: $$x(t)= \begin{cases} 16 \quad & ; -7<t<0 \\ 0 & ; \text{otherwise} \\ \end{cases}$$ And the unit impulse ...
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### Solution to $\frac{d^n y}{dx^n} = y$ for integer $n$

In short, my question is about the solution of $\frac{d^n y}{dx^n} = y$ for positive integer $n$, where $y$ is real. For example, if $n = 2$, the solution is $y = c_1e^x+c_2e^{-x}$. The work that I ...
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### Obtaining function from integrals

I have a question I cannot figure out if it's correct or not. I have a function $f(x,y)$ unknown which I'd like to obtain. Starting from $q(y) = \int dxf(x,y)$ and $g(x) = \int dyf(x,y)$ I think I ...
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### Differential equation of a LRC circuit as Voltage output/voltage input?

So in this presentation i have got, there is a LRC circuit, and they get this differential equation, but there is no procedure explained how they have got it. This is basicially a series LRC circuit ...
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### Solution of the following differential equation .

$2x^3ydy + (1-y^2)(x^2y^2 -1)dx =0$ The correct solution to the problem is : $x^2y^2 = (Cx - 1)(1 - y^2)$ Where C is a constant .
$$x'= \frac{-x}{t}$$ So I'm trying to solve the above equation in the following method. We can write $x'= \frac{dx}{dt}$. this makes $$\frac{dx}{dt} = \frac{-x}{t}$$  \frac{1}{x}dx = \frac{-1}{t}...